Scheduling is the operational core because the business is sold in time slots. What you need is booking that shows real availability, handles rescheduling without a conversation, sends reminders automatically, and enforces your cancellation terms. No shows are the largest silent cost in tutoring, and automated reminders plus a stated cancellation policy applied consistently address most of it. The policy matters more than the software: a booking system with no consequence for late cancellation will not change behaviour.
Payment should be collected in advance or on a recurring basis rather than session by session. Packages paid upfront, or a monthly subscription, remove the collection problem and improve retention because a family that has paid for a block turns up. If you take money in advance, remember that it is not yet earned revenue and should be treated accordingly, both in your accounting and preferably in where it is held.
Delivery technology depends on whether sessions are remote, and if they are, reliability outweighs features considerably. A stable connection, a decent microphone, and a shared whiteboard cover most subjects. Recording sessions is worth considering because students revisit them, and it requires explicit permission, which for a minor means from the parent rather than the student.
Communication with minors is where the requirements differ from other businesses. Parents should be able to see how you are contacting their child, communication should happen through channels that produce a record rather than through personal messaging, and your policies should say so. This protects the student and it protects you, and it is the kind of thing that is straightforward to establish at the start and awkward to introduce later.
Progress records are the retention mechanism and the underused one. A short written note after each session, and a periodic summary showing what has changed, gives a parent evidence that the money is producing something. Tutoring is frequently cancelled not because it stopped working but because nobody could tell whether it was working, and a simple record addresses that directly.
Keep the parent as the account holder even where the student is the one attending, because that determines who receives the invoices, who can change the booking, and who consents to anything requiring permission. Businesses that set up accounts in the student name discover the complication at the first billing dispute or the first request for a recording, and correcting it afterward means recreating records rather than changing a setting.